Systematics 1. Introduction

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3. Systematics 1. Introduction

Systematics is a process in which some p r o p e r t y of the lanthanides, and sometimes the c o m p l e t e rare earth group, is examined in detail to see h o w it varies f r o m one r a r e earth element to another. In general one examines the group as a whole, or at least a p o r t i o n of the group (at least three m e m b e r s ) to see the overall trend and a n o m a l i e s f r o m the trend. F o r the lanthanides the a t o m i c n u m b e r is usually the variable, b u t size is frequently used. F o r the rare earth group, which includes Sc and Y in a d d i t i o n to the lanthanides, the a t o m i c or ionic size is the only logical variable o n which the c o m p a r i s o n is based.

A s we will see systematics is a powerful tool. It gives us insight to and u n d e r s t a n d - ing of the chemical and physical natures of the rare earth metals, alloys and

compounds. It can be used to check the reliability of experimental results, i.e., to choose between two conflicting pieces of data or to question a result because it does not follow the trend established within the expected experimental error. It can also be used to predict unknown properties, especially by interpolation between known data. The use of systematics in evaluating RM binary phase diagrams (where R is rare earth and M is a non-rare earth metal) and RM crystallographic data have been described by Gsehneidner and Calderwood (1983).

The smooth systematic variation of the physical properties of the lanthanides was recognized as soon as the first physical property measurements had been made on the entire group of lanthanide elements. The first such measurements were the lattice parameters of the rare earth oxides from which the ionic radii were deduced. In 1924, Goldschmidt,while compiling the ionic radii of the elements, noted that those of the lanthanide elements decreased in a regular and smooth fashion from La to Lu, and coined the words "lanthanide contraction" to describe this systematic trend (e.g.

see Goldschmidt et al., 1925, von Hevesy, 1927 and Goldschmidt, 1954). He recognized that this contraction is due to the increased effective nuclear charge as an additional 4f electron is added as one proceeds along the lanthanide series in increasing atomic number. Since then, scientists have utilized systematics and deviations from the general trends to understand the natures and behaviors of lan.thanide materials. For example, deviations in a plot of the metallic radii vs. the atomic number of the lanthanide elements indicate that Eu and Yb are divalent because of their much larger radii (see Beaudry and Gschneidner, 1978), and that aCe has a valence of - 3.7 because its radius lies below the established curve (see Koskenmaki and Gschneidner, 1978).

Properties that depend only upon the outer valence electrons will show similar regular variations when plotted against the atomic number. These include, in addition to the lattice constants of compounds, alloys, etc., the density, unit cell volume, melting point and most thermodynamic properties. Other properties, such as the magnetic susceptibility, moments, ordering temperatures, etc. and optical spectra, are due to the 4f electrons and thus the variations of these properties depend upon the S, L and J quantum numbers. These properties are not used in our systematic analysis of rare earth binary phase diagrams. Another set of related properties also behave anomalously--the boiling point, heat of sublimation at 298 K, the cohesive energy and any other thermodynamic quantity that involves the heat of sublimation in calculating a thermodynamic cycle, such as a Born-Haber cycle. The reason for this anomaly is that the boiling point, heat of sublimation, etc.

not only involve the solids (which have three 6s5d valence electrons and a 4f n configuration) but also the gaseous atoms (which have a 4fn+16s 2 configuration except for La, Ce, Gd and Lu, which have 4f n6s25d configurations). The tendency to change electron configurations during evaporation or sublimation gives rise to a sawtooth-like variation in a plot of the property vs. atomic number (Beaudry and Gschneidner, 1978).

Yttrium and scandium data can also be examined along with those of the lanthanides. Generally the values for yttrium fall between those of dysprosium and

154 K.A. GSCHNEIDNER and F.W. CALDERWOOD

holmium, but this is not always true and some care and discretion must be used in the analysis. The data for scandium can be compared with those of the lanthanides

• /

by plotting the results as a ~unction of the metallic radius, but again much care and discretion must be used if the scandium data deviate from the expected trends.

3.2. Early work

Work on correlating the crystal structure sequence in the pure metals and intra rare earth alloys started in about the mid-1960s, shortly after the discovery by Spedding et al. (1962) of the formation of the Sm-type structure in alloys between a light lanthanide metal and a h e a r lanthanide (or yttrium) metal. The next major impetus was provided by the high pressure work of Bell Telephone Laboratory's research group (Jayaraman and Sherwood, 1964a, b; McWhan and Bond, 1964) who found that under pressure hcp Gd transformed to the Sm-type structure; Sm (rhombohedral, Sm-type structure) transformed to the dhcp structure; and dhcp La transformed to the fcc structure. Thus it became evident that the crystal structure sequence in the lanthanide series at 1 atm pressure varied from fcc ~ dhcp ---, Sm- type(or 8) ~ hcp, while pressure caused the reverse structure sequence.

The occurrence of these phases, especially the Sm-type structure, was discussed by a variety of investigators who tried to correlate these behaviors to c / a ratlos (Spedding et al., 1962; Harris et al., 1966), size (Nachman et al., 1963), atomic volume (McWhan and Bond, 1964), 4f hybridization with conduction electrons (Gschneidner and Valletta, 1968; R.H. Langley, 1981), the average atomic number (Harris and Raynor, 1969), and the relative occupancy of s and d orbitals (Duthie and Pettifor, 1977; Skriver, 1983). Of these factors, all except size (and atomic volume) are valid and can be used with some modification to correlate the structure occurrences at atmospheric and high pressures. In the mid-1970s Johansson and Rosengren (1975) developed a generalized phase diagram for the rare earth elements by superimposing the individual pressure-temperature diagrams on one another by connecting their melting points and in some instances their hcp-bcc phase transformation temperatures. From this, one could correlate the general trends of the various phase transformations, but their generalized diagram is ditticult to use, especially for binary alloys, because the temperature and pressure axes are displaced from one lanthanide to another.

3.3. Generalized phase diagram at 1 atm

Recently, by making use of systematics, Gschneidner (1985a) proposed a generalized phase diagram for the trivalent intralanthanide and yttrium-lanthanide binary alloys. This single diagram, shown in fig. 121a, represents a total of 91 phase diagrams. The diagram was constructed from the melting and transformation temperatures of the trivalent lanthanides, assuming Ce and Sm are trivalent. The data for Eu and Yb in this figure are for the hypothetical trivalent states. The various phase boundaries were estimated Dom the various intralanthanide phase diagrams. In order to accomplish this, each lanthanide was assigned a value called

the "systematization n u m b e r " (SN), which is shown o n the top of the diagram. It is seen that SN varies from zero for La to 14 for Lu, and is equal to the total number of 4f electrons for the respective trivalent lanthanide element. However, as Gschneidner (1985a)points out, this coincidence is fortuitous and no implication was intended that these two numbers are related. Yttrium was assigned an SN value of 9.5.

In order to estimate the location of a critical composition or phase boundary, x B (mole fraction of B), in any intralanthanide alloy system between lanthanides A and B o n e uses the lever law

(1 - x » ) S N A + x s S N » = SNcp

and the systematization n u m b e r of the critical point or phase boundary, SNcp. In the above equation SN A and SN B a r e the respective systematization numbers of A and B. T h e SNcp v a l u e s are listed in table 5.

The phase boundaries shown in fig. 121a are shown as narrow lines, but in the actual phase diagrams the two phase regions are generaUy between 2 and 10 at%

wide; for neighboring lanthanides they may be as wide as 20at%. Gschneidner (1985a) believed that the boundaries can be estimated to within 5 at% and the temperatures to within _+ 25°C.

T h e dip at Ce in the melting point and two tränsformation points is due to the valence fluctuation character of Ce, and presents no problem when Ce is one of the end-members of a binary phase diagram. ~ h e n Ce and R a r e alloyed the ap- proprfate phases and the corresponding temperatures will be found as indicated in the generalized phase diagram.

This dip, however, is ignored in phase diagrams in which La is an end-member and the other end-member is any other lanthanide (or yttrium) except Ce. Since this dip is due to the 4f electron in Ce, a feature that cannot be mimicked by a pseudo-Ce composition made up by a L a - R alloy with SN = 1, no dip is to be expected, and none is found in the known L a - R systems (e.g., see section 2.4.1, fig.

5). F o r a L a - R diagram, the melting points of La and R are connected with a straight line allowing for a liquidus-solidus separation at intermediate compositions.

A shallow dip, however, is expected in the bcc -~ hcp transformation since this is due TABLE 5

Systematization number for phase boundary limits or critical points on the generalized phase diagram.

Phase boundary Temperature Systematization

limit or (°C) nurnber

critical point

fcc-dhcp > 400 1.2

dhcp-8(Sm) 25 4.4

8(Sm) decomposition Maximum 5.0

6(Sm)-hcp 25 5.6

bcc-hcp-liquid > 1200 9.8

156 K.A. GSCHNEIDNER and F.W. CALDERWOOD

to a eutectoid decomposition of the bcc phase to the dhcp plus fcc phases at SN = 1 (see section 2.4.1, fig. 5). For Y, the hcp-bcc and bcc-liquid transformation temperatures are about 50°C higher than the values shown on the generalized phase diagram at SN = 9.5. Thus the respective phase boundaries must be prorated upward accordingly when one is calculating a particular Y - R diagram.

The continuous phase transition from the dhcp phase at low SN values ( < 5) to the hcp phase at high SN values ( > 5) as shown in fig. 12la is consistent with our current understanding of binary intra rare earth alloys involving a light lanthanide metal (0 < SN < 4) with a heavy lanthanide or yttrium metal (7 < SN _< 14). This has been discussed in section 2.1.4.

Although the generalized phase diagram may be used with confidence for most binary combinations, it appears it may not be valid with regard to the formation of the 8 phase in lutetium alloys with the light lanthanides. On the basis of an X-ray study in both the La-Lu (section 2.8.1) and N d - L u (section 2.36.1) systems, no

&phase structure was observed in the X-ray patterns of alloys that were expected to have this phase.

An analysis of the generalized binary alloy phase diagram showed that there are only 13 possible types of diagrams that can be formed (Gschneidner 1985a). Of the known phase diagrams, 11 of the 13 types have been observed experimentally. Using the scheme outline above, Gschneidner calculated the hypothetical diagrams of the other two types, La-Er, which represents type 5, and Sm-Ho, which represents type 11. A type 5 phase diagram is formed between one component having three polymorphic phases and a second monophasic component that is not isomorphic with any phase of the first component, but a region of complete miscibility is formed between the dhcp phase of the first component and hcp phase of the second component. A type 11 phase diagram is formed between Sm, which has three polymorphic forms, and the heavy lanthanides, which have only the hcp structure (Ho, Er, Tm and Lu).

3.4. The high pressure generalized phase diagram

In a second paper Gschneidner (1985b) extended the atmospheric generalized intra rare earth binary phase diagram to high pressures: 1, 2 and 4GPa (see figs.

121b, c and d, respectively). The high pressure diagrams are almost entirely based on high pressure behaviors of the pure metals. The greatest uncertainty in these diagrams lies in the intermediate temperature range between room temperature and the hcp-bcc and bcc-liquid phase boundaries.

Examination of Fig. 121 shows that the fcc and Sm-type phase fields tend to expand at the expense of the dhcp- and hcp-phase fields, respectively, as the pressure is increased, and since the fcc-phase field is growing faster than the Sm-type phase field, the fcc structure is expected to be the stable phase at sufficiently high pressures.

The formation of the collapsed aCe phase from -/Ce is evident in figs. 121b and c.

At higher pressure the transformation of the fcc aCe to the «U-type structure is noted in fig. 121d. The influence of these dense phases on the melting point is quite evident as the Ce dip becomes much more severe as the pressure increases.

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Fig. 123. The 25°C isothermal section of the intra rare earth generalized binary phase diagram from 0 to 24 G P a (0 to 240 kbar). The solid circles are transformation points observed in the pure metals and the × data points are taken from the phase boundaries given in the generalized phase diagrams shown in fig.

121, The dashed line in the fcc region indicates the formation of the distorted fcc or triple hexagonal close-packed (thcp) phase from the normal fcc phase with increasing pressure. The letters " T B C " mean tetragonal body-centered.

T h e pressure dependence of the bcc region is quite interesting: The region disappears with increasing pressure for La, Ce and Pr but grows with increasing pressure from Sm (SN = 5) to the end of the lanthanide seiles Lu, SN = 14. For Nd the bcc region expands as the pressure increases to 2 G P a and then contracts as more pressure is applied. Gschneidner (1985b) correlated this behavior of the bcc region wi'th the d occupation number n« (see table 6). Based on the disappearance of TLa (bcc) with pressure and the termination of the bcc phase at 1 atm between Dy and Ho, Gschneidner estimated that the bcc phase is stable over the range 1.6 > n d _> 2.2.

Since n« increases with increasing pressure by about 0 . 1 / G P a (Skriver, 1983), application of pressure causes the light lanthanides, which have high n d values, to exceed the 2.2 upper limit and disappear, while for those elements which have small n a values ( < 1.6), pressure increases n« so that it exceeds the 1.6 lower limit causing the bcc to form.

In order to determine the structure(s) of a given binary rare earth alloy system, one follows the procedures outlined in section 3.3 utilizing the appropriate isopiestic section shown in fig. 121. Unfortunately, it is difficult to make such estimates for pressures between the four isopiestic sections. In order to improve such estimations a set of eight isothermal sections at 200, 400, 600, 700, 800, 1000, 1200 and 1500°C is presented in fig. 122, where the ordinate extends up to 4 GPa. Since little if any high temperature data exist above 4 G P a the diagrams were cut off at this pressure. At room temperature, however, data extend up to higher pressures as shown in fig. 123.

With increasing temperature the aCe phase (seen in fig. 122a) disappears (absent in fig. 122b) because of the a - T critical point at 327°C. T h e fcc phase is seen to grow at the expense of the dhcp as the temperature is raised (figs. 122a, b and c), but the formation of the bcc phase interrupts this process (fig. 122d). The bcc phase continues to expand with increasing temperature eliminating the dhcp between 1000 and 1200°C (figs. 122f and g) and also the Sm-type phase between 1200 and 1500°C (figs. 122g and h). The Sm-type phase region expands with increasing pressure and maintains its width of about three SN values with increasing temperature until about

TABLE 6

The d occupation number n« for the trivalent rare earth metals (after Skriver, 1983).

R n d R t/d

La 1.99 Dy 1.64

Ce 1.97 Ho 1.60

Pr 1.93 Er 1.56

Nd 1.89 Tm 1.52

Pm 1.85 Yb a 1.48 b

Sm 1.81 Lu 1.44

Eu a 1.77 b Sc 1.58

Gd 1.72 Y 1.60

Tb 1.68

~Hypothetical trivalent R.

bEstimated by reviewers.

158 K.A. G S C H N E I D N E R and F.W. C A L D E R W O O D

1000°C (figs. 122a-f) and it begins to narrow down and disappears as noted above.

But at low pressures ( < 0.5 GPa) this phase, which is about one SN value wide (fig.

122a and b) begins to narrow at 600°C (fig. 122c) and disappear between 700 and 800°C (figs. 122d and e). At high temperature the liquid phase becomes important for the light lanthanides at practically all pressures.

The extremely high pressure behavior of the lanthanides at room temperature, as shown in fig. 123, is quite interesting. The fcc and Sm-type phase fields expand with increasing pressure at the expense of the dhcp- and hcp-phase fields, respectively.

The formation of the unusual aU-type structure for Ce and Pr and the body-centered tetragonal (tbc) phase for Ce is evidence for delocalization of the 4f electron in these metals at high pressure (Koskenmaki and Gschneidner, 1978; Grosshans et al., 1983;

Benedict et al., 1984).

The dashed line in fig. 123 in the fcc region represents the formation of a ùdistorted fcc" phase from the true fcc polymorph with increasing pressure (Grosshans et al., 1982). More recently, Smith and Akella (1984) suggested that this new phase has a triple hexagonal close-packed (thcp) structure, which had been predicted by McMahan and Young (1984).

The rapid rise of the hcp-Sm-type phase boundary beyond Tm (fig. 123), and the anomalous high pressures of transformation in Y (hcp ~ Sm-type, Sm-type ~ dhcp and dhcp ~ fcc) relative to lanthanide elements was cited by Gschneidner (1985b) as evidence for 4f valence electron hybridization having a significant role in determining the lanthanide crystal structure. He noted, however, that d occupation number probably is more important in determining which crystal structure would form, as had been proposed by others (e.g., see Duthie and Pettifor, 1977; Skriver, 1983).

3.5. Lattice spacings and Vegard's law

Lattice spacings have been reported for about 40% of the possible intra rare earth systems, and these have been summarized in earlier portions of this review. Several interesting trends have been noted by Gschneidner (1985a) and these are discussed below.

For systems wherein both end-members have the same structure at room temperature, either positive deviations from Vegard's law are found in either the a (25 systems) o r c (10 systems) lattice parameter (e.g., see section 2.2.2 and fig. 2), or no deviations (22 systems) are observed. This behavior is consistent with the second order elasticity model of Gschneidner and Vineyard (1962) that predicts only positive deviations from Vegard's law regardless of the solute to solvent size ratio.

Other models predict both positive and negative deviations from Vegard's law (Gschneidner and Vineyard, 1962).

When the end-members have different room temperature structures q u i t e a different behavior is found. For a light lanthanide metal as a solvent the a parameter generally exhibits a negative deviation from Vegard's law, while the c parameter shows a positive deviation (e.g., see section 2.4.2 and figs. 6 and 7, respectively). This behavior was found in 28 of 33 systems for both the a and c lattice parameters. But if the solvent is a heavy lanthanide or Y or Sc, the a parameter deviates positively

from Vegard's law (15 of 18 systems) and the c parameter tends to exhibit a negative deviation (7 of 18 systems) or no deviation (7 of 18 systems). Typical examples can be found in section 2.38.2, figs. 67 and 68, respectively.

Gschneidner (1984a) found that these behaviors can be understood in terms of the d occupation numbers, which are smaller for the heavy lanthanides, Y and Sc than those of the light lanthanides (see table 6). It is known that an increase in the amount of d character results in a smaller

c/a

ratio (Legvold et al., 1977).

Gschneidner (1984a) argued that when a heavy lanthanide (or Y or Sc) is added to a fight lanthanide the amount of d character seen by the heavy lanthanide solute is larger than in the pure heavy lanthanide metal. This "excess" d character can be reduced by increasing the

c/a

ratio beyond that expected from a simple addition of atomic volumes and this leads to a negative deviation from Vegard's law in the a parameter of the mätrix and also a positive deviation in the c parameter, which is the case when the dhcp and Sm phases are solvents. The opposite occurs when a light lanthanide is dissolved in a heavy lanthanide matrix: The light lanthanide sees a smaller amount of d character and this deficiency will be ameliorated by decreas- ing the

c/a

ratio. This causes a positive deviation from Vegard's law of the a lattice parameter and a negative deviation of the c lattice parameter, which is observed for a hcp solvent containing light lanthanide solutes.

Of eourse in addition to the dependence on the d occupation number, second order elasticity effects also apply. This probably accounts for the fact that for the heavy lanthanide solvents the departures from Vegard's law exhibited in the c lattice spacing are only slightly negative or do not occur because the electronic and second order elasticity effects tend to cancel one another.

Thus, the variation of lattice parameters in the intra rare earth alloys is reasonably weil understood in terms of elasticity theory and electronic configurations (d occupation numbers). Exceptions to the above correlations are probably due to poor experimental data.

3.6.

Thermodynamics

Lundin and Yamamoto (1967) investigated the P r - N d , S m - G d , Sm-Y and G d - Y systems to further the understanding of alloy formation. Variables under eonsideration were electronic structure, crystal structure, atomic diameter, valences and electronegativity. The P r - N d system (see section 2.25) represented a system in which there were no significant differences in these parameters and their investiga- tion revealed thermodynamic ideality in this system (see section 2.25.3, fig. 45). The S m - G d system (section 2.39) had one major variable, the crystal structure Sm-type vs. hcp, and showed large negative deviations from ideality (see section 2.39.3, fig.

71), which were attributed to lattice strain. The G d - Y system (section 2.52), in which the crystal structures are the same, had a large difference in electronic structure between the two eomponents but was found to be thermodynamically ideal (see section 2.52.4, fig. 91). The Sm-Y system (section 2.40) varies from the other systems in that the room temperature crystal structures (Sm-type vs. hcp) and the electronic structures [(6s5d)34f s vs. (5s4d) 3] of the components are different while

Dalam dokumen Handbook on the Physics and Chemistry of Rare Earths (Halaman 158-173)

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3. Systematics 1. Introduction

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